Obstruction theory

In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants. In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent vectors. Obstruction theory turns out to be an application of cohomology theory to the problem of constructing a cross-section of a bundle. The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. It is traditionally called Eilenberg obstruction theory, after Samuel Eilenberg. It involves cohomology groups with coefficients in homotopy groups to define obstructions to extensions. For example, with a mapping from a simplicial complex X to another, Y, defined initially on the 0-skeleton of X (the vertices of X), an extension to the 1-skeleton will be possible whenever the image of the 0-skeleton will belong to the same path-connected component of Y. Extending from the 1-skeleton to the 2-skeleton means defining the mapping on each solid triangle from X, given the mapping already defined on its boundary edges. Likewise, then extending the mapping to the 3-skeleton involves extending the mapping to each solid 3-simplex of X, given the mapping already defined on its boundary. At some point, say extending the mapping from the (n-1)-skeleton of X to the n-skeleton of X, this procedure might be impossible. In that case, one can assign to each n-simplex the homotopy class πn-1(Y) of the mapping already defined on its boundary, (at least one of which will be non-zero). These assignments define an n-cochain with coefficients in πn-1(Y). Amazingly, this cochain turns out to be a cocycle and so defines a cohomology class in the nth cohomology group of X with coefficients in πn-1(Y). When this cohomology class is equal to 0, it turns out that the mapping may be modified within its homotopy class on the (n-1)-skeleton of X so that the mapping may be extended to the n-skeleton of X.